NOTES 5 DYNAMICS OF A RIGID ROTOR FLUID FILM BEARING SYSTEMrotorlab.tamu.edu/me626/Notes_pdf/Notes05 rigid rotor on JBs 10.pdf · notes 5. dynamics of a rigid rotor-fluid film bearing

Notes 5. DYNAMICS OF A RIGID ROTOR-FLUID FILM BEARING SYSTEM. Dr. Luis San Andrés © 2010 1

NOTES 5 DYNAMICS OF A RIGID ROTOR-FLUID FILM BEARING SYSTEM Lecture 5 restates the analysis for static equilibrium in a journal bearing. Next, it considers the dynamics of the simplest rigid rotor bearing system supported on journal bearings. For small amplitude journal motions about an equilibrium position, the analysis proceeds to linearize the fluid film forces and introduces the concepts of bearing force coefficients, namely, 4 stiffnesses, 4 damping and 4 inertia coefficients. Formulas for the direct and cross-coupled stiffness and damping coefficients of a short length journal bearing are derived. The analysis focuses on the stability of the rigid rotor-bearing system to determine the threshold rotor speed at which the system loses its equivalent damping and develops ever growing motions at a whirl frequency that coincides with the rotor-bearing system natural frequency. The low and high magnitudes of the Sommerfeld number show whereas the system operates stably or not. The ½ whirl frequency ratio reveals a typical stability limit of lubricated journal bearings. The effect of rotor flexibility on further reducing the threshold speed of instability is noted since the rotor-bearing natural frequency is also lowered. An appendix provides a physical explanation of the follower force, induced by the cross-coupled stiffnesses, that drives the rotor bearing system into whirl. Remedies to avoid or delay the instability are given. Actual examples of instabilities and measurements in the author’s laboratory make evident the harmful, potentially catastrophic, whirl instability. A list of industrial or commercial bearing configurations with noted advantages and disadvantages complements the lecture. Nomenclature

C Radial clearance [m] Cij Bearing damping force coefficients, i,j=X,Y [N.s/m] D=2R Bearing diameter e Journal eccentricity [m]

,e eφ Vr, Vt . Journal center radial and tangential velocities [m/s] F Fluid film force acting on journal surface [N] . 2 2 2 2

X Y r tF F F F F= + = + Fo ½ Static load [N] h Film thickness. H=h/c Kij Bearing stiffness force coefficients, i,j=X,Y [N/m] Ke Bearing equivalent stiffness [N/m] Krot Elastic rotor stiffness (one side) [N/m] L Bearing axial length M ½ Mass of rigid rotor [kg] Mij Bearing added mass force coefficients, i,j=X,Y [N.s2/m] mc=ps

2 Dimensionless critical mass P Hydrodynamic pressure [Pa] R Bearing radius [m] r, t Moving coordinate system S Sommerfeld number t Time (s) u Mass imbalance [kg]


X, Y Inertial coordinates system ΔX, ΔY Small amplitude displacements in (X,Y) coordinate system Δe,eo Δφ Small amplitude displacements in (r,t) coordinate system ε e/C. Journal eccentricity ratio δ u/C. imbalance parameter ρ Fluid density [kg/m3] σ 2

4 o

LR LF C

μ Ω ⎛ ⎞⎜ ⎟⎝ ⎠

. Modified Sommerfeld number

μ Absolute viscosity [N.s/m2]Γ o rotF K . Static (elastic) sag at rotor midspan [m]φ Journal attitude angle Θ, θ Circumferential coordinates ω Characteristic whirl frequency [rad/s] ωn (Keq/M)½ . Rotor-bearing system natural frequency [rad/s] Ω Journal rotational speed [rad/s] Ωs Threshold speed of instability [rad/s] Subscripts

a Ambient value o Static or equilibrium condition s, fs Threshold of instability for rigid and flexible rotor XX,XY,YX,YY Indices of force coefficients in fixed (X,Y) coordinate system rr,rt,tr,tt Indices of force coefficients in moving (r,t) coordinate system


Equations of motion of a rigid rotor supported on plain journal bearings Consider, as shown in Fig. 5.1, a symmetric rigid rotor of mass 2M that carries a static load (2Fo=W) along the X axis. Two identical plain journal bearings support the rotor. The equations of motion of the rotating system at constant rotational speed Ω are given by:

2

2

sin( )

cos( )X o

Y

M X F M u t F

M Y F M u t

= + Ω Ω +

= + Ω Ω (5.1)

where u is the magnitude of the imbalance vector, X(t) and Y(t) are the coordinates of the rotor mass center, and (FX, FY)are the fluid film bearing reaction forces.

Since the rotor is rigid, the center of mass displacements are identical to those of the journal bearing centers, i.e.

)()(),()( tetYtetX YX == (5.2) The rotor-bearing static equilibrium is defined by

OOYXYoX eeeFFFOOOO

φ,or,,0, ⇒=−= (5.3) where (eo ,φo) are the static equilibrium journal eccentricity and attitude angle, respectively. The static fluid film reaction force components are such that:

X

Y

2Fo

bearing

Rotor (journal)fluid film

JournalRotation (Ω)

Disk 2M

Journal bearing

X

Y

2Fo

disk

Clearance circle

Ωt

e

Static load

u Rigid shaft

Figure 5.1. Rigid rotor supported on journal bearings. (u) imbalance, (e) journal eccentricity


OtOrYY

OtOrXoXo

OOOO

OOOO

FFFF

FFFFFF

φφ

φφ

cossin00

sincos0

+==⇒=

−==⇒=+ (5.4)

Recall that 2 2 2 2X Y r tF F F F F= + = +

At equilibrium, the region of positive fluid film pressure extends from θ1 = 0 to θ2 = π. In a short length journal bearing, the radial and tangential components of the static fluid film force Fo are

( ) ( ) 2322

3

22

2

2

3

14;

1 ε

επμ

ε

εμ

−

⋅Ω+=

−

Ω−=

CLRF

CLRF

OO tr

(5.5)

where R= D/2, L and C are the journal radius, axial length and radial clearance, respectively. ε=eo/C is the journal center eccentricity ratio, ε<1.0; μ is the lubricant absolute viscosity, and Ω=(rpm π/30) is the rotor speed in rad/s. Figure 2 depicts the force components, radial and tangential, growing rapidly (nonlinearly) with the journal eccentricity e/C.

0 0.2 0.4 0.6 0.8 1100

1 .103

1 .104

1 .105

-FrFt

Static Forces for short length bearing

journal eccentricity (e/C)

Rad

ial a

nd T

ange

ntia

l for

ces [

N]

*

Note that the short length bearing forces are proportional to the lubricant viscosity and rotor surface speed (ΩR), the bearing length (L3), and inversely proportional to the radial clearance (C2). Most importantly, the bearing forces grow rapidly (non-linearly) with the journal eccentricity (ε=e/C). Each bearing reaction force balances a fraction of the applied static load Fo = ½ W for a symmetric rotor bearing system. Thus,

( ) ( )( )

2 2 221 22 222

16 1

4 1O Oo r t

LF F F R LC

ε π εεμε

+ −⎛ ⎞= + = Ω ⎜ ⎟⎝ ⎠ −

(5.6)

X

Y

r

t

Fo -Fro

Fto

φo

Figure 5.2. Radial and tangential forces in short length journal bearing μ=0.019 Pa.s, L=0.05 m, C=0.1 mm, Speed 3, 000 rpm L/D=0.25


The equilibrium attitude angle (φo) between the static load direction and the eccentricity vector is

( )( )21

tan4

o

o

to

r

FF

π εφ

ε

−= − =

⋅ (5.7)

Note that as ε → 0, φo → ½ π (journal eccentricity is perpendicular to the static load direction), whereas ε → 1, φo → 0 (journal eccentricity parallel or aligned to load direction). The bearing design parameter is the modified Sommerfeld number (σ)

( )( ) 222

222

116

14 επεε

εσμ

−+

−==⎟

⎠⎞

⎜⎝⎛Ω

CL

FLR

o

(5.8)

For a rated operating condition, σ is known since the bearing geometry, speed, fluid type (viscosity) and load are known. Then Eqn. (5.8) gives a relationship to determine (iteratively) the equilibrium eccentricity ratio, ε =e/c, that generates the film reaction force balancing the applied static load Fo. Recall that, Large Sommerfeld (σ) numbers (small load W, high speed Ω, large lubricant viscosity μ) determine small operating journal eccentricities or nearly centered operation, i.e. ε → 0.0 and attitude angles approaching 90°; and Small Sommerfeld (σ) numbers (large load W, low speed Ω, light lubricant viscosity μ) determine large operating eccentricities, i.e. ε→1.0 and attitude angle approaching 0° Figures 4.6-8 in Lecture 4 depict the Sommerfeld number and attitude angle versus the journal eccentricity and the locus of the journal center within the clearance circle. The same figures are reproduced here in a smaller format.

0.01 0.1 1 1000.10.20.30.40.50.60.70.80.9

1

Sommerfeld number

jour

nal e

ccen

trici

ty e

/c

ε

σ0.01 0.1 1 100

102030405060708090

Sommerfeld number

Atti

tude

ang

le (d

eg)

φ

σ Fig. Eccentricity ratio (ε) and attitude angle (φ) versus Sommerfeld number (σ) in a short journal bearing


Consider, as represented in Figure 5.3, small amplitude journal motions about the equilibrium position. These motions are defined as

)(),( teeeteee YYYXXX OO

Δ+=Δ+= (5.9.a) or

)(),( tYYYtXXX OO Δ+=Δ+= (5.9.b) or conversely,

)()(),()( ttteete OO φΔφφΔ +=+= (5.9.c) with

YYXX

YYXX

eedt

Ydeedt

Xd

eedtdYee

dtdX

Δ==Δ==

Δ==Δ==

2

2

2

2

.

. (5.10)

The journal dynamic displacements in the (r, t) coordinate system are related to those in the (X,Y) fixed system by the linear transformation

⎥⎦

⎤⎢⎣

⎡Δ

Δ⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡ΔΔ

)()(

cossinsincos

tete

ee

OOO

OO

Y

X

φφφφφ

(5.11)

Similar relationships hold for the journal center velocities and accelerations.

Fo φo

eΔφ

Δe

Δe

Δe

eYo

eXo

eo

t

r

Y

clearance circle

Figure 5.3. Small amplitude journal motions about a static equilibrium position

X


Note that the small amplitude motions assumption means Cee YX <<ΔΔ , , i.e., the journal dynamic displacements are much smaller than the bearing clearance. The fluid film forces are general functions of the journal center displacements and velocities, i.e.

YXteteteteFF YXYX ,)],(),(),(),([ == ααα (5.12) The assumption of small amplitude motions about an equilibrium position allows expressing the bearing reaction forces as a Taylor Series expansion around the static journal position (eXo, eYo), i.e.

O

O

X X X X X XX X

Y Y Y Y Y YY Y

F F F F F FF F X Y X Y X YX Y X Y X Y

F F F F F FF F X Y X Y X YX Y X Y X Y

∂ ∂ ∂ ∂ ∂ ∂= + Δ + Δ + Δ + Δ + Δ + Δ

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂= + Δ + Δ + Δ + Δ + Δ + Δ

∂ ∂ ∂ ∂ ∂ ∂

(5.13)

Definition of dynamic force coefficients in fluid film bearings Fluid film bearing stiffness (Kij) ij=X,Y , damping (Cij) ij=X,Y and inertia force coefficients are defined as

;j

iij X

FK

∂∂

−=j

iij X

FC

∂∂

−= ; ;j

iij X

FM

∂∂

−= i,j=X,Y (5.14)

For example, KXY = -∂FX/∂Y corresponds to a stiffness produced by a fluid force in the X direction due to a journal static displacement in the Y direction. By definition, this coefficient is evaluated at the equilibrium position with other journal center displacements and velocities equal to zero. The negative sign in the definition assures that a positive magnitude stiffness coefficient corresponds to a restorative force. The coefficients (KXX, KYY) are known as the direct stiffness terms, while the coefficients (KXY, KYX) are referred as cross-coupled. Figure 5.4 provides an idealized representation of the bearing force coefficients as mechanical parameters.

Fluid inertia or added mass coefficients ;j

iij X

FM

∂∂

−= i,j=X,Y where YX , are journal center

accelerations. Fluid inertia coefficients are of particular importance in superlaminar and turbulent flow bearings and seals handling liquids (large density). The inertia force coefficients or apparent masses have a sound physical interpretation and are always present in a fluid film bearing. Inertia coefficients are of large magnitude especially in dense liquids. However, the effect of inertia forces on the dynamic response of rotor-bearing systems is only of importance at


large excitation frequencies (This fact also holds for most mechanical systems subjected to fast transient motions).

Note that the defined force coefficients allow the representation of the dynamic fluid film bearing (or seal) forces in terms of fundamental mechanical parameters K, C, M. However, this does not mean that these coefficients must be accordance with customary knowledge. For example, the “viscous” damping coefficients may be negative, i.e. non-dissipative, or the stiffness coefficients non restorative or non conservative. Fluid film force coefficients in the radial and tangential directions (r, t) are also defined. Thus, the radial and tangential fluid film forces are expressed as (stiffness and damping for simplicity)

φφ

φφ

φφ

Δ−Δ−Δ−Δ−=

Δ∂

∂+Δ

∂∂

+Δ∂

∂+Δ

∂∂

+=

ortrrortrrro

oo

rro

o

rrror

eCeCeKeKF

ee

Fee

Fee

Fee

FFF (5.15a)

φφ

φφ

φφ

Δ−Δ−Δ−Δ−=

Δ∂

∂+Δ

∂∂

+Δ∂

∂+Δ

∂∂

+=

otttrotttrto

oo

tto

o

tttot

eCeCeKeKF

ee

Fe

eF

ee

Fe

eF

FF (5.15b)

Note that φΔΔ oee, are the journal center radial and tangential (small) velocities in the (r, t) coordinate system, respectively.

KXX, CXX

journal

bearing

X

Y

KXY, CXY

KYX, CYX

KYY CYY Kij = - ΔFi/ΔXj

Cij = -ΔFi/d(ΔXj)/dt

Figure 5.4. The “physical” representation of dynamic force coefficients in fluid film bearings


The relationship between the force coefficients in both coordinate systems is easily determined from equation (5.11) as:

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡

oo

oo

tttr

rtrr

oo

oo

YYYX

XYXX

oo

oo

tttr

rtrr

oo

oo

YYYX

XYXX

CCCC

CCCC

KKKK

KKKK

φφφφ

φφφφ

φφφφ

φφφφ

cossinsincos

cossinsincos

cossinsincos

cossinsincos

(5.16)

Substitution of the force coefficient definitions (5.14) into equation (5.13) gives the following

⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

Δ⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛ΔΔ

⎥⎦

⎤⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎟⎟

⎠

⎞⎜⎜⎝

⎛

YX

CCCC

YX

KKKK

F

F

tFtF

YYYX

XYXX

YYYX

XYXX

Y

X

Y

X

O

O

)()(

(5.17)

And, the governing equations of motion for the rigid-rotor-bearing system, Eqn. (5.1) become

⎟⎟⎠

⎞⎜⎜⎝

⎛ΩΩ

Ω=⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎥⎦

⎤⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝

⎛

Δ

Δ⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛

ΔΔ

⎥⎦

⎤⎢⎣

⎡tt

uMYX

KKKK

YX

CCCC

YX

MOOM

YYYX

XYXX

YYYX

XYXX

cossin2 (5.18)

where FXo = Fo= ½W and FYo = 0. These differential equations are linear and represent the rotor-bearing system dynamics for small amplitude motions about the equilibrium position. Fluid inertia effects are altogether neglected in the traditional stability analysis of rotor-lubricated bearing systems. Force coefficients for the short length journal bearing The general definition of fluid film bearing dynamic force coefficients is above. The analytical derivation of these coefficients for the short length journal bearing follows. The film thickness for an aligned cylindrical journal bearing is

cos ;h C e(t) ( )θ θ ϕ= + = Θ − (5.19)

For small amplitude motions about the equilibrium position, )()();()( ttteete oo φφφ Δ+=Δ+= , where Δe and Δφ are small radial and angular displacement quantities, respectively. Eqn. (5.19) is rewritten with θ =Θ -φo as

( ) ,sinsincoscos φθφθ Δ+ΔΔ++= eeCh o


and, for small amplitude motions note that cos(Δφ)~1, sin(Δφ) ~Δφ. Then neglecting second order terms,

θφθθ sincoscos Δ+Δ++= oo eeeCh = 1oh h+ (5.20) where,

θφθθ sincos;cos 10 Δ+Δ=+= oo eeheCh (5.21) are the zeroth-order and first-order or perturbed film thicknesses, respectively. Recall that the Reynolds equation for the short length journal bearing model is1:

θμ ∂∂Ω

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ h

th

zPh

z 212

3

(5.22)

and, θ

θθφθ

θθ

θφθ

sin;cossin

sincos1

oo

oo

o

eh

eehh

eet

hth

−=∂∂

Δ+Δ−∂∂

=∂∂

Δ+Δ=∂∂

=∂∂

(5.23)

Substitution of (5.23) into (5.22) gives:

θφθφμ

sin22

cos212

3

⎭⎬⎫

⎩⎨⎧ Ω

Δ−⎥⎦⎤

⎢⎣⎡ Ω

−Δ+⎭⎬⎫

⎩⎨⎧ Δ

Ω+Δ=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ eeee

zPh

z oo (5.24)

Integration of Eqn. (5.24) in the axial direction and applying the boundary conditions at the sides of the bearing, i.e. P = Pa at z = ± ½ L, gives:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎥

⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧ Ω

Δ−⎥⎦⎤

⎢⎣⎡ Ω

−Δ+⎭⎬⎫

⎩⎨⎧ Δ

Ω+Δ=−

4sin

22cos

26 2

233

LzeeeeHC

PP ooa θφθφμ (5.25)

where H = h/C. Integration of the pressure field on the journal surface gives the radial and tangential components of the fluid film force, i.e.,

( ) dzdRtzPFF

L

t

r θθθ

θ

πθ

θ

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ ∫∫=

=

sincos

,,2

2

1 0

2

0

(5.26)

1 This equation is valid for (L/D)<<0.50 and incompressible, isoviscous lubricants. No thermal effects are accounted for in this simple form of the classical Reynolds equation.


where the positive (uncavitated) pressure region lies between θ1 = 0 and θ2 = π when Pa is set as zero (nil). Note that it is assumed the perturbed pressure field, due to small amplitude journal motions about the equilibrium position (eo, φo), does not affect the extent of the steady state lubricant cavitation region, i.e. from 0 to π. This assumption is clearly void if the motions are large in character. By the way, the concept of linear force coefficients is also inadequate when motion amplitudes are large. Substitution of Eqn. (5.25) into Eqn. (5.26) and integration in the axial direction renders

θθθθφθφμ

π

dH

eeH

eeC

LRFF

t

r⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧ Ω

Δ−⎥⎦⎤

⎢⎣⎡ Ω

−Δ+⎭⎬⎫

⎩⎨⎧ Δ

Ω+Δ=⎟⎟

⎠

⎞⎜⎜⎝

⎛− ∫ sin

cossin22

cos2 3030

0

3

3

(5.27)

However, the cubic term in the denominator (H3) also depends on the perturbed journal center displacements. A first-order Taylor series expansion of this terms gives for h/C=H

14

03

03 3 hhhh −−− −= (5.28)

where θφθθ sincos;cos 10 Δ+Δ=+= oo eeheCh . Substitution of Eqn. (5.28) into (5.27) and neglecting second-order terms, i.e. products of small quantities such as Δe ·Δφ, etc., gives after some considerable algebraic manipulation

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎥⎦

⎤⎢⎣

⎡

++−++−Ω

−⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡φεε

εεμφ

μe

eJJJJJJJJ

CLR

ee

JJJJ

CLR

FF

FF

t

r

t

r30

411

321

420

3

214

023

124

113

3

3

203

113

113

023

3

3

0

0

3333

2

(5.29)

where θθθπθ

θ

dH

J i

jkkji

00

)(cos)(sin2

1

∫=

=

= are definite integrals and Ho =( 1 + ε cos θ ).

The bearing stiffness and damping force coefficients are, from Eqn. (5.29), specified as


3

23 3r11 12

3 4rr 33 2

3 3t11 30

tt 3 4 23 3 2o

302 21

rt 3 43

2 ( 1 + ) RL RL F = - + 3 = = - K J J2 eC ( 1 - )C RL RL F = + 3 = = - J JK 2 ( 1 - )C C e

RL = + 3 =J JK 2C

μ μ ε εεε

μ μ εεϕε

μ ε

Ω Ω ∂∂

Ω Ω ∂∂

Ω

3r

3/ 23 2o

23 3t20 21

tr 3 4 5/ 23 3 2

RL F = - 4 ( 1 - )C e

( 1 + 2 )RL RL F = - + 3 = - = - J JK 2 e4 ( 1 - )C C

μ πϕε

μ μ π εεε

Ω ∂∂

Ω Ω ∂∂

(5.30)

eF - =

eF - = C =

) - 1 () 2 - (

CRL = J

CRL = C

eF - =

) - 1 ( 2

CRL = J

CRL = C

eF - =

) - 1 () 2 + 1 (

2

CRL = J

CRL = C

t

o

rtr223

31133

3

rt

o

t3/22

32033

3

tt

r5/22

2

3

30233

3

rr

∂∂

∂∂

∂∂

∂∂

φε

εμμ

φε

πμμ

εεπμμ

3 (5.31)

The ( φΔΔ e ,e o

) correspond to the journal center radial and tangential velocities in the (r, t) coordinate system, respectively. Note that the stiffness coefficients (Kij) ij=r,t are proportional to the rotational speed (Ω) and fluid viscosity (μ). The damping coefficients (Cij) ij=r,t are not a a direct function of the angular speed but depend only on the fluid viscosity and the journal equilibrium position. Without journal rotation there cannot be a fluid film bearing stiffness. Dimensionless Force Coefficients The literature presents the force coefficients in dimensionless form according to

00

;F

CCcFCKk ijijijij

Ω== i,j=X,Y (5.32)

where Fo is the static load applied on each bearing (in the X direction). [Note that the total load W=2Fo is shared by the two bearings in a symmetric rotor mount].

Recall that ( )σ

μ4/ 2 RLCLFo

Ω= , where (σ) is the modified Sommerfeld Number defined as (See

Notes 4)


( )

( ) 222

222

116

14 επεε

εσμ

−+

−==⎟

⎠⎞

⎜⎝⎛Ω

CL

FLR

o

(4.8)

Using the following definitions:

( ) ( ) 23222

2

1sin;

14cos

ε

πσεφε

σεφ−

==+=−

==−= oo

totoo

o

roro F

Ff

FF

f (5.33)

the dimensionless force coefficients in the (r, t) coordinate system become,

( )( )

( )( )

( )( )

εε

εεε

εε

εεε

εεε

2;1

;1

21

2;11

212;112

2

2

2

2

2

2

tottrott

totr

rorttrtort

torrrorr

fcfk

fk

fccfk

fcfk

==

−+

−=

−===

−+

=−+

=

(5.34)

Force coefficients in the (X,Y) coordinate system are easily obtained using the matrix transformation Eqn. (5.16). After a lengthy algebraic procedure,

( )

( )

( )

( ) 222

222

222

222

211

11

11

211

εεε

εεε

εεε

εεε

++−

==

+−−

==

−+−

==

++−

==

roto

oXYXY

roto

oYXYX

toro

oYYYY

roro

oXXXX

ff

FCKk

ff

FCKk

ff

FCKk

ff

FCKk

( ) ( )

( ) ( )

( ) ( ) YXtro

oXYXY

tto

oYYYY

rot

oXXXX

cff

FCCc

ff

FCCc

ff

FCCc

o

o

o

=+−+−⋅

=Ω

=

+−+−

=Ω

=

−++−

⋅=

Ω=

2222

2222

2222

1212

1212

1212

εεεε

εεεε

εεεε

(5.35) recall that the X-direction is along the static load Fo. Figures 5.5 and 5.6 depict the dimensionless force coefficients, stiffness and damping, as functions of both the journal eccentricity and the modified Sommerfeld number (σ), respectively. Both representations are necessary since sometimes the journal eccentricity is known a priori


while most often, the design parameter, i.e. the Sommerfeld number, is known in advance. In general, the physical magnitude of the stiffness and damping coefficients increases rapidly (nonlinearly) with the journal eccentricity (load too!). Note that the dimensionless force coefficients do not represent the actual physical trends. For example, at eo=0, KXX=KYY=0, but the dimensionless values kXX=kYY=0 in the figures show a definite value. This peculiar result follows from the definition of dimensionless force coefficients using the applied load (Fo). Thus, as eo→0 , the bearing load Fo is also nil.


0 0.2 0.4 0.6 0.8 10.1

1

10

KxxKyyKxy-Kyxk.eq

journal eccentricity (e/c)

Stiff

ness

*

0 0.2 0.4 0.6 0.8 11

10

100

cxxcyycxycyx

journal eccentricity (e/c)

Dam

ping

*

Figure 5.5. Short length journal bearing (dimensionless) stiffness and damping force coefficients vs. journal eccentricity (ε)

ε

ε High speed Low load Large viscosity Low speed

Large load Low viscosity


0.01 0.1 1 100.1

1

10

kxxkyykxy-kyxkeq

Sommerfeld #

Stiff

ness

*

0.01 0.1 1 101

10

100

CxxCyyCxyCyx

Sommerfeld Number

Dam

ping

*

Figure 5.6. Short length journal bearing (dimensionless) stiffness and damping force coefficients vs. modified Sommerfeld number (σ)

σ

σ High speed Low load Large viscosity

Low speed Large load Low viscosity

2

4⎟⎠⎞

⎜⎝⎛Ω

=cL

WRLμσ


Dynamic force Coefficients for journal centered operation, i.e. static load=0 As the journal center approaches the bearing center, eo→0, and from the formulas,

3 3

3 3

0

;4 2 2

rr tt rt tr

rt tr tt rr

K K C C

R L R Lk K K c c C CC C

μ π μ π

= = = =

Ω Ω= = − = = = = =

(5.36)

At e→0, φo = 90o, so the force coefficients in the (X,Y) system are given as:

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

−+

=⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡

−⎥⎦

⎤⎢⎣

⎡+

−=⎥

⎦

⎤⎢⎣

⎡

cc

cc

CCCC

kk

kk

KKKK

YYYX

XYXX

YYYX

XYXX

00

0110

00

0110

00

0110

00

0110

Hence

2;

24 3

3

3

3 πμπμC

LRcCCcC

LRkKK YYXXYXXY ===Ω

=Ω

==−= (5.37)

Thus, at the centered journal position the bearing offers no direct (support) stiffness but only cross-coupled support. A small static load applied on the bearing will cause a journal displacement in a direction orthogonal (perpendicular) to the load. This phenomenon is found in nearly all fluid film bearings of rigid geometry.


Stability analysis of rigid rotor supported on plain journal bearings For small amplitude journal motions about the equilibrium position (eo, φo), the equations of motion of a rigid rotor supported on (linear) fluid bearings are:

⎟⎟⎠

⎞⎜⎜⎝

⎛ΩΩ

Ω=⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛

ΔΔ

⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛

ΔΔ

⎥⎦

⎤⎢⎣

⎡tt

uMYX

KKKK

YX

CCCC

YX

MOOM

YYYX

XYXX

YYYX

XYXX

cossin2 (5.38)

Introduce the dimensionless variables:

Cut

CYy

CXx =Ω=

Δ=Δ

Δ=Δ δτ ,,, (5.39)

where C is the bearing radial clearance and Ω is the journal or rotor speed (regarded as invariant). Substitution of Eqn. (5.39) into (5.38) gives:

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ΔΔ

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡′Δ

′Δ⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡′′Δ

′′Δ)cos()sin(22

ττ

δpyx

kkkk

yx

cccc

yx

pYYYX

XYXX

YYYX

XYXX (5.40)

where F

MC = pdd =

o

22 Ω;)( '

τis a dimensionless mass, and kij = Kij (C/Fo), cij = Cij (CΩ /Fo)

are the dimensionless dynamic force coefficients. It is of interest to study if the rotor-bearing system is stable for small amplitude journal center motions (perturbations) about the equilibrium position. To this end, set the imbalance parameter δ = 0 in the equations above to obtain,

Ω

F

Non-rotating structure

F

Rotating structure

F F

The significance of cross-coupling effect from fluid film journal bearings


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ΔΔ

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡′Δ

′Δ⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡′′Δ

′′Δ002

yx

kkkk

yx

cccc

yx

pYYYX

XYXX

YYYX

XYXX (5.41)

If the rotor-bearing system is to become unstable, this will occur at a threshold speed of rotation (Ωs) and the rotor will perform (undamped2) orbital motions at a whirl frequency (ωs). These motions, satisfying equation (5.42), are of the form:

1;; −===== jeBeByeAeAx jtjjtj ss τωωτωω (5.42)

where ss Ωωω = is known as the whirl frequency ratio, i.e. the ratio between the rotor whirl or precessional frequency and the rotor onset speed of instability. Substitution of Eqn. (5.42) into (5.41) leads to:

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

++−++++−

00

22

22

BA

cjkpcjkcjkcjkp

YYsYYssYXsYX

XYsXYXXsXXss

ωωωωωω

(5.43)

The determinant of the system of equations must be zero for a non-trivial solution of the homogenous system of equations, i.e.

( ) ( )( ) ( ) 0

2222

=+⋅+−++−⋅++−=

XYsXYYXsYX

YYsYYssXXsXXss

cjkcjkcjkpcjkp

ωωωωωωΔ

(5.44)

After a rather lengthy algebraic manipulation, the real and imaginary parts of Δ above render,

o

S

YYXX

YXXYXYYXXXYYYYXXeqss F

MCcc

kckcckckkp2

22 ωω =

+−−+

== (5.45)

and

( )( ) 22

⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

=−

⋅−−−=

s

s

YXXYYYXX

YXXYYYeqXXeqs cccc

kkkkkk ωω (5.46)

For a given value of journal eccentricity (εo), i.e. a given Sommerfeld number (σ ), one evaluates Eqn. (5.45) to obtain the dimensionless equivalent stiffness keq, and then (5.46) to obtain the whirl frequency ratio Sω . This substitution then yields 22

Seqs kp ω= (system critical mass) which in turn renders the onset speed of instability Ωs.

2 Recall that in a second order mechanical system an equivalent damping ratio>0 causes the damping or attenuation of motions induced by small perturbations. A damping equal to zero produces sustained periodic motions without decay or growth and indicates the threshold between stability and instability (amplitude growing motions).


Figures 5.7 and 5.8 depict the whirl frequency ratio ss Ωωω = and the dimensionless threshold speed of instability (ps) versus both the journal eccentricity and Sommerfeld number, respectively. Note that for near centered journal operation, i.e. large Sommerfeld numbers, the whirl frequency is 0.50, i.e. half-synchronous whirl. Other important information is also obtained. If one assumes that the current (operating) rotational speed Ω is the onset speed of instability, then from the relations above, the magnitude of ½ system mass (M) is obtained, and which would make the rotor-bearing system become unstable. This mass is known as the critical mass, Mc, and corresponds to the limit mass which the system can carry dynamically. If the total mass is equal or larger than twice Mc, then the system will be unstable at the rated speed Ω (3). The whirl frequency ratio, s sω ω= Ω , is the ratio between the rotor whirl frequency and the onset speed of instability. Note that this ratio, as given by Eqn. (5.46), depends only on the fluid film bearing characteristics and the equilibrium eccentricity, and it is independent of the rotor characteristics (rotor mass and flexibility). The parameter keq is a journal bearing (dimensionless) equivalent stiffness and depicted in Figures 5.5 and 5.6. From the definitions of threshold speed and whirl ratio, ( )oss FCMp 22 Ω= and sss Ωωω = , then

eqo

eqs KCF

kM =⎟⎠⎞

⎜⎝⎛=2ω

Thus, the whirl or precessional frequency is given by

eqs n

KM

ω ω= = (5.47)

i.e., the whirl frequency equals the natural frequency of the rigid rotor supported on journal bearings. For operation close to the concentric position, εo → 0, i.e. large Sommerfeld numbers (no load condition), the force coefficients are, see Eqn. (5.37),

0;;;0 ==−==== YXXYYXXYYYXXYYXX cckkcckk (5.37)

( ) XXXYXYXXXXeq ckcckk += =0 and

0.50 as 0s XY

s XX

kc

ω ε= = →Ω

(5.48)

3 Recall that each bearing carries half the static load, and also half the dynamic or inertia load (2.McC Ω2).


The 0.5 magnitude for whirl frequency ratio (WFR) (or 50% whirl as is called in industry) is a characteristic of hydrodynamic plain journal bearings. It shows us that at the onset of instability the rotor whirls at its natural frequency, which equals to 50% of the rotor speed. Furthermore, under no externally applied loads, Fo=0, as in vertically turbomachinery, the bearing possesses no support stiffness, i.e. Keq=0 and the system natural frequency (ωn) is zero, i.e. the rotor-bearing system whirls at all speeds. Note that if kXY = 0, i.e. the fluid film bearing does not show cross-coupled effects, then the WFR = 0, i.e. no whirl occurs and the system is ALWAYS stable. (Asymmetrical) cross-coupled stiffnesses are thus responsible for the instabilities so commonly observed in rotors mounted on journal bearings. If the whirl frequency ratio is 0.50, then the maximum rotational speed that the rotor-bearing system can attain is just,

max 2 20.50

ss n

ω ω ωΩ = = = (5.49)

i.e., twice or two times the natural frequency (or observed rigid rotor critical speed). Figures 5.7, 5.8, and 5.9 show, respectively, the whirl frequency ratio, the dimensionless critical mass parameter (ps), and the dimensionless critical mass (ps)2 versus the Sommerfeld number and operating journal eccentricity. The results show that a rigid-rotor supported on plain journal bearings is always STABLE for operation with journal eccentricity ratios ε > 0.75 (small Sommerfeld numbers) for all L/D ratios. Note that the critical mass and the whirl ratio are relatively insensitive for operation with eccentricities εo < 0.50. Keep in mind that increasing the rotational speed of the rotor-bearing system determines larger Sommerfeld numbers, and consequently, operation at smaller journal eccentricities for the same applied static load. Thus, operation at ever increasing speeds will eventually lead to a rotor dynamically unstable system as the results show. Effects of Rotor Flexibility A similar analysis can be performed considering rotor flexibility. This analysis is more laborious though straightforward. The analysis shows that the whirl frequency ratio is not affected by the rotor flexibility. However, the onset speed of instability decreases dramatically! The relationship for the threshold speed of instability of a flexible rotor is:

22

1

ssf

eq

ppk

C

=Γ⎛ ⎞+ ⎜ ⎟

⎝ ⎠

(5.50)

where the sub index f denotes the flexible rotor, Krot is the rotor stiffness on each side of the center disk, and o rotF KΓ = is the rotor static sag or elastic deformation at midspan.

bearing 2M KRot


The elastic shaft and bearing are mounted in series, i.e. the bearing and shaft flexibilities add (reciprocal of stiffnesses), and thus the equivalent system stiffness is lower than that of the bearings alone, and therefore the system natural frequency is lower. Figure 5.10 shows the threshold speed of instability (psf) for a flexible rotor mounted on plain short length journal bearings. Note that the more flexible the rotor is, the lower the threshold speed of instability. If the fluid film bearings are designed too stiff (small Sommerfeld numbers), then the natural frequency of the rotor-bearing system is just (Krot/M)0.5, irrespective of the bearing configuration. Postcript See the Appendix to these notes for further understanding on the nature of the cross-coupled coefficients driving the whirl motion. The MATHCAD programs attached include the algebraic formulas for evaluation of the bearing force coefficients in actual applications.


\

0.01 0.1 1 1000.10.20.30.40.50.60.70.80.9

1

S#

whi

rl fr

eque

ncy

ratio

.5

* *Sommerfeld # (σ)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.9

1

e/c

whi

rl fr

eque

ncy

ratio

.5

*Eccentricity (e/c)

Figure 5.7. Whirl frequency ratio vs. modified Sommerfeld number (σ) and journal eccentricity (ε)

High speed Low load Large viscosity


2

4⎟⎠⎞

⎜⎝⎛Ω

=cL

WRLμσ


0.01 0.1 1 100123456789

10

S#* Sommerfeld # (σ)

unstableunstable

stablestable

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10123456789

10

e/c*Eccentricity (e/c)

stablestable

unstableunstable

Ps = M Ωs2 c/Fo

Figure 5.8. Dimensionless threshold speed of instability (ps) vs. modified Sommerfeld number (σ) and journal eccentricity (ε)



2

4⎟⎠⎞

⎜⎝⎛Ω

=cL

WRLμσ


0.01 0.1 1 100123456789

10

S#* Sommerfeld # (σ)

unstableunstable

stablestable

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10123456789

10

e/c*Eccentricity (e/c)

stablestable

Figure 5.9. Dimensionless critical mass (mc=ps2) vs. modified Sommerfeld number

(σ) and journal eccentricity (ε).




0.01 0.1 1 100

2

4

6

rigidT/c=0.1T/c=1T/c=10

Threshold speed (ps) for flexible rotor

Modified Sommerfeld number

Thre

shol

d sp

eed

(ps)

low speed high speedlarge load <--- ---> small loadlow viscosity large viscosity

Figure 5.10. Dimensionless threshold speed of instability (ps) for flexible rotor vs. modified Sommerfeld number (σ). Static sag (Γ/c) varies

bearing 2M KRot

unstable

Rotor more flexible

Sommerfeld # (σ)


References consulted [1] Turbomachinery Rotordynamics, (chapter 3), D. Childs, John Wiley & Sons, Inc., 1993. [2] Rotordynamics of Turbomachinery, J.M. Vance, J., Wiley Inter-Science Pubs., 1988. [3] Self-Excited, Stationary Whirl Orbits of a Journal in a Sleeve Bearing, J. Lund, Ph.D.

Thesis, Rensselaer Polytechnic Institute, Troy, N.Y., 1966. [4] The Stability of an Elastic Rotor in Journal Bearings with Flexible Supports, J, Lund,

ASME Journal of Applied Mechanics, pp. 911-920, 1965. [5] A Table of the Journal Bearing Integrals, J.F. Booker, ASME Journal of Basic Engineering,

pp. 533-535, 1965. [6] Tribology – Friction, Lubrication & Wear, A. Szeri, Hemisphere Pubs, 1980.

μR 2.5 10 6−⋅ 6894.757⋅

N s⋅

m2⋅:= fluid viscosity at reference

etmperature: TR 50:= α 0.030:=

Oil viscosity-temperature coefficient (1/deg C)

T 60:= degC operating (feed) temperature

kshaft 108 Nm⋅:= 1/2 shaft stiffness a 0.2 c⋅:= Amplitude of imbalance on rotor disk

Static shaft deflection due to rotor weight = % of clearance δ

Wkshaft

:= 1N 1 kg m s-2=

Γδ

c:=

Γ 0.222=

δ 1.334 10 5−× m= if Γ > 1 then rotor is quite flexible

μ μR eα− T TR−( )⋅

⋅:= μ 0.013N s⋅

m2= Operating viscosity

Static performance

Stability and Imbalance Response of a Jeffcott-RotorSupported on Short Length Journal Bearings

(c) Dr. Luis San Andres UT/2000, TAMU/2006 Extended with eigen analysis: 10/00 TAMU

Disk 2M

Jou abearing

X

Y

2Fo

disk

Clearance circle

Ωt

e

Static load

uRigid shaft

g g pp j g(u) imbalance, (e) journal eccentricity

DATA for rotor WT 600 4.448⋅ N⋅:= Rotor weight Gravitational acceleration g 9.807 m s-2=

WWT

2:= W 1.334 103

× N= Static load per bearing

MWg

:= M 136.071 kg= 1/2 rotor mass

BEARING GEOMETRY, OIL viscosity and Operating conditions: Top shaft speed for analysisD 0.15 m⋅:= journal diameter

RPM_max 12000:=

L 2 0.025⋅ m⋅:= bearing length

c 0.060 mm⋅:= radial clearance

Journal eccentricity ratio and attitude angle for STATIC equilibriumShort bearing solution for an isoviscous-incompressible lubricant

2 4 6 8 10 120

0.5

1

rotor speed (kRPM)

jour

nal e

ccen

trici

ty e

/c

Y

0

30

6090

120

150

180

210

240270

300

330

0.8

0.6

0.4

0.2

0

X

0 2 4 6 8 10 120

90

rotor speed (kRPM)

Atti

tude

ang

le (d

eg)

Power loss (kW) / bearing

0 5 10 150

25

50

rotor speed (kRPM)

Pow

er (k

W) p

er b

earin

g

Side flow rate / bearing (LPM)

0 2 4 6 8 10 120.2

0.4

0.6

0.8

rotor speed (kRPM)

Bea

ring

flow

rate

(LPM

)

JB force coeffs & stability

Stiffness and damping force coefficients vs shaft speed

6 120

1 .108

2 .108

3 .108

4 .108

5 .108

6 .108

KxxKyyKxy-Kyx

rotor speed (kRPM)

Stiff

ness

(N/m

)

0 5 10 150

10

20

30

40

50

CxxCyyCxyCyx

rotor speed (kRPM)

Dam

ping

(N.s/

m)

0 2 4 6 8 10 123.5 .107

4 .107

4.5 .107

rotor speed (kRPM)

Equi

v. S

tiffn

ess (

N/m

)

Equivalent bearing stiffness for rigid rotor

Whirl frequency ratio

0 2 4 6 8 10 120.4

0.45

0.5

0.55

rotor speed (kRPM)

whi

rl fr

eque

ncy

ratio

Threshold speed of instability(rpm)

0 2 4 6 8 10 12

6

12

Threshold speedShaft SpeedSyst nat frequency

Rigid Rotor

rotor speed (kRPM)

Thre

shol

d sp

eed

(kR

PM)

Critical Rotor Mass for rigid rotor Recall rotor mass: 2 M⋅ 272.142 kg=

0 2 4 6 8 10 120.1

1

10

100

rotor speed (kRPM)

Mcr

titic

al/R

otor

Mas

s

System goes unstable when ratio < 1

Eigenvalues rig rotor

0 2 4 6 8 10 120.5

0

0.5

1

rotor speed (kRPM)

Dam

ping

ratio

damping ratio for system,

Search for intersection of real part with Y=0 axis to determinethrshold speed of instability.

0 2 4 6 8 10 120

5

10

15

nat freq(Keq/M)^0.51X

rotor speed (kRPM)

Nat

ural

freq

uenc

y an

d ro

tor s

peed natural frequency - from

eigenvalue analysis and from (Keq/M) formula

Damping ratio of flex rotor system

0 2 4 6 8 10 120.5

0

0.5

1

rotor speed (kRPM)

Dam

ping

ratio

Eigenvalues: flexible rotor

The threshold speed of instability is lower for the flexible rotor than for the rigid rotor model.

Γ 0.222=

0 2 4 6 8 10 120

5

10

15

flex rotorrigid rotornat. freq rotsync line

Flexible Rotor

rotor speed (kRPM)

Thre

shol

d sp

eed

(kR

PM)

ωnnWFRn Ω tfn

⋅:=

Natural Frequency (rpm) of Flexible Shaft

NEARLY RIGID ROTORthreshold speed of instability Ω tfn

Ωtn

1 keqnΓ⋅+( )

1

2

:=

δ 1.334 10 5−× m=Γ 0.222=Static Deflection (Γ=δ/c) and Stiffness of Flexible shaft

kshaft 1 108×

Nm

=Effect of shaft flexibility on threshold speed of instability:

Synchronous response

y yc cos ω t⋅( )⋅ ys sin ω t⋅( )⋅+=x xc cos ω t⋅( )⋅ xs sin ω t⋅( )⋅+=

Y Yc cos ω t⋅( )⋅ Ys sin ω t⋅( )⋅+=X Xc cos ω t⋅( )⋅ Xs sin ω t⋅( )⋅+=

The rotor disk (X,Y) and journal center displacements (x,y) are synchronous with the imbalance excitation, i.e.

MASSLESS BEARINGS ROTOR

ac

0.2=

md2 Y⋅

dt2

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅ kshaft Y y−( )⋅+ m a⋅ ω2

⋅ sin ω t⋅( )⋅=

imbalance displacementCI J,( )dxdt

dydt

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

⋅ KI J,( )x

y⎛⎜⎝⎞⎟⎠

⋅+ k−x X−

y Y−⎛⎜⎝

⎞⎟⎠

⋅=

a 1.2 10 5−× m=

md2 X⋅

dt2

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅ kshaft X x−( )⋅+ m a⋅ ω2

⋅ cos ω t⋅( )⋅=

The equations of motion for both rotor and journal bearings are given below. The coordinates of rotor and disk motion have their origin at the static equilibrium position. No damping at rotor midspan, no mass lumped at the bearings.

Synchronous imbalance response of flexible rotor

RPMωelastic 8.186 103×=

Natural frequency - flex rotor and simple formula (rigid rotor)

0 2 4 6 8 10 120

5

10

15


rotor speed (kRPM)

Nat

ural

freq

uenc

y an

d ro

tor s

peed

0 2 4 6 8 10 120

2 .10 5

4 .10 5

6 .10 5

X rotorY rotor

Imbalance response Mid disk

rotor speed (kRPM)

Am

plitu

des -

roto

ramplitudes in [m]

Amplitudes of motionat rotor midspan

0 2 4 6 8 10 120

7.5 .10 6

1.5 .10 5

X journalY journal

Imbalance response bearing location

rotor speed (kRPM)

Am

plitu

des -

jour

nal

c 6 10 5−× m=

a 1.2 10 5−× m=

Amplitudes of motionat bearings

Exercise: Calculate the major and minor axes of the ellipses describing the (X,Y) motions. See Appendix A of Childs' Rotordynamics Book:.

Notes: You could update this program to account for a) bearing mass MB, a fraction of total rotor mass,b) introduce damping at the rotor midspan, Cs.

Damping ratio of flex rotor system

0 2 4 6 8 10 120.5

0

0.5

1

rotor speed (kRPM)

Dam

ping

ratio

Eigenvalues: flexible rotor

The threshold speed of instability is lower for the flexible rotor than for the rigid rotor model.

Γ 2.224=

0 2 4 6 8 10 120

5

10

15

flex rotorrigid rotornat. freq rotsync line

Flexible Rotor

rotor speed (kRPM)

Thre

shol

d sp

eed

(kR

PM)

ωnnWFRn Ω tfn

⋅:=

Natural Frequency (rpm) of Flexible Shaft

FLEXIBLE ROTORthreshold speed of instability Ω tfn

Ωtn

1 keqnΓ⋅+( )

1

2

:=

δ 1.334 10 4−× m=Γ 2.224=Static Deflection (Γ=δ/c) and Stiffness of Flexible shaft

kshaft 1 107×

Nm

=Effect of shaft flexibility on threshold speed of instability:

Synchronous response

y yc cos ω t⋅( )⋅ ys sin ω t⋅( )⋅+=x xc cos ω t⋅( )⋅ xs sin ω t⋅( )⋅+=

Y Yc cos ω t⋅( )⋅ Ys sin ω t⋅( )⋅+=X Xc cos ω t⋅( )⋅ Xs sin ω t⋅( )⋅+=

The rotor disk (X,Y) and journal center displacements (x,y) are synchronous with the imbalance excitation, i.e.

MASSLESS BEARINGS ROTOR

ac

0.2=

md2 Y⋅

dt2

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅ kshaft Y y−( )⋅+ m a⋅ ω2

⋅ sin ω t⋅( )⋅=

imbalance displacementCI J,( )dxdt

dydt

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

⋅ KI J,( )x

y⎛⎜⎝⎞⎟⎠

⋅+ k−x X−

y Y−⎛⎜⎝

⎞⎟⎠

⋅=

a 1.2 10 5−× m=

md2 X⋅

dt2

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅ kshaft X x−( )⋅+ m a⋅ ω2

⋅ cos ω t⋅( )⋅=

The equations of motion for both rotor and journal bearings are given below. The coordinates of rotor and disk motion have their origin at the static equilibrium position. No damping at rotor midspan, no mass lumped at the bearings.

Synchronous imbalance response of flexible rotor

RPMωelastic 2.589 103×=

Natural frequency - flex rotor and simple formula (rigid rotor)

0 2 4 6 8 10 120

5

10

15


rotor speed (kRPM)

Nat

ural

freq

uenc

y an

d ro

tor s

peed

0 2 4 6 8 10 120

7.5 .10 5

1.5 .10 4

X rotorY rotor

Imbalance response Mid disk

rotor speed (kRPM)

Am

plitu

des -

roto

ramplitudes in [m]

Amplitudes of motionat rotor midspan

0 2 4 6 8 10 120

5 .10 6

1 .10 5

X journalY journal

Imbalance response bearing location

rotor speed (kRPM)

Am

plitu

des -

jour

nal

c 6 10 5−× m=

a 1.2 10 5−× m=

Amplitudes of motionat bearings

Exercise: Calculate the major and minor axes of the ellipses describing the (X,Y) motions. See Appendix A of Childs' Rotordynamics Book:.

Notes: You could update this program to account for a) bearing mass MB, a fraction of total rotor mass,b) introduce damping at the rotor midspan, Cs.

Notes 5. APPENDIX B. TYPES OF JOURNAL BEARINGS. Dr. Luis San Andrés © 2009 1

NOTES 5. APPENDIX B OTHER TYPES OF LUBRICATED JOURNAL BEARINGS Compressors, turbines, pumps, electric motors, electric generators and other rotating machines

are commonly supported on fluid film bearings. In the past, the vast majority of these bearings

were plain journal bearings. As machines have achieved higher speeds, rotor dynamic instability

problems such as oil whirl have brought the need for other types of bearing configurations.

Cutting axial grooves in the bearing to provide a different oil flow pattern across the lubricated

surface generates some of these geometries. Other bearing types have various patterns of variable

clearance (preload and offset) to create a pad film thickness that has strongly converging and

diverging regions, thus generating a direct stiffness for operation even at the journal centered

position. Various other geometries have evolved as well, such as the tilting pad bearings which

allow each pad to pivot, and thus to take its own equilibrium position. This usually results in a

strongly converging film region for each loaded pad and the near absence of cross-coupled

stiffness coefficients.

TYPES OF HYDRODYNAMIC BEARINGS:

The Tables below list in a condensed form some of the advantages and disadvantages of various

practical bearing configurations.

Other Bearing types

Cutting axial grooves in the bearing to supply oil flow into the lubricated surfaces generates some of these geometries.

Other bearing types have various patterns of variable clearance (preload and offset) to create a pad film thickness that has strongly converging wedge, thus generating a direct stiffness for operation even at the journal centered position.

In tilting pad bearings, each pad is able to pivot, enabling its own equilibrium position. This feature results in a strongly converging film region for each loaded pad and the near absence of cross-coupled stiffness coefficients.

Commercial rotating machinery implements bearing configurations aiming to reduce and even eliminate the potential of hydrodynamic instability (sub synchronous whirl)

OTHER BEARING GEOMETRIES

Used primarily on high speed turbochargers for PV and CV engines

1. Subject to oil whirl (two whirl frequencies from inner and outer films (50% shaft speed, 50% [shaft + ring] speeds)

1. Relatively easy to make2. Low Cost

Floating Ring

Round bearings are nearly always “crushed” to make elliptical or multi-lobe

1. Subject to oil whirl1. Easy to make2. Low Cost

Axial Groove

Bearing used only on rather old machines

1. Poor vibration resistance2. Oil supply not easily contained

1. Easy to make2. Low Cost3. Low horsepower loss

Partial Arc

Round bearings are nearly always “crushed” to make elliptical bearings

1. Most prone to oil whirl1. Easy to make2. Low Cost

Plain Journal

Comments Disadvantages Advantages Bearing Type

Table Fixed Pad Non-Pre Loaded Journal Bearings


Currently used by some manufacturers as a standard bearing design

1. Expensive to make properly2. Subject to whirl at high speeds

1. Good suppression of whirl2. Overall good performance3. Moderate cost

Three and Four Lobe

High horizontal stiffness and low vertical stiffness -may become popular - used outside U.S.

1. Fair suppression of whirl at moderate speeds2. Load direction must be known

1. Excellent suppression of whirl at high speeds2. Low Cost3. Easy to make

Offset Half (With Horizontal Split)

Probably most widely used bearing at low or moderate rotor speeds

1. Subject to oil whirl at high speeds2. Load direction must be known

1. Easy to make2. Low Cost3. Good damping at critical speeds

EllipticalComments Disadvantages Advantages Bearing Type

Fixed Pad Pre-Loaded Journal Bearings Table 2


Fixed Pad Pre-Loaded & Hydrostatic Bearings Table

Generally high stiffness properties used for high precision rotors

1. Poor damping at critical speeds2. Requires careful design3. Requires high pressure lubricant supply

1. Good suppression of oil whirl2. Wide range of design parameters3. Moderate cost

Hydrostatic

Used as standard design by some manufacturers

1. Complex bearing requiring detailed analysis2. May not suppress whirl due to non bearing causes

1. Dams are relatively easy to place in existing bearings2. Good suppression of whirl3. Relatively low cost 4. Good overall performance

Multi-Dam Axial Groove or Multiple-Lobe

Very popular in the petrochemical industry. Easy to convert elliptical over to pressure dam

1. Goes unstable with little warning2. Dam may be subject to wear or build up over time3. Load direction must be known

1. Good suppression of whirl2. Low cost3. Good damping at critical speeds4. Easy to make

Pressure Dam (Single Dam)

CommentsDisadvantagesAdvantagesBearing Type


Tilting Pad Bearings & Foil BearingsTable

Used mainly for low load support on high speed machinery (APU units).

1. High cost.2. Dynamic performance not well known for heavily loaded machinery.3. Prone to subsynchronous whirl

1.Tolerance to misalignment.2.Oil-free

Foil bearing

Widely used bearing to stabilize machines with subsynchronous non-bearing related excitations

1. High Cost2. Requires careful design3. Poor damping at critical speeds4. Hard to determine actual clearances5. Load direction must be known

1. Will not cause whirl (no cross coupling)

Tilting Pad journal bearing

Flexure pivot, tilting pad bearing

CommentsDisadvantagesAdvantagesBearing Type

Bump foils

Top foil

Spot weld

Journal

Bearing sleeve

Notes 5. APPENDIX B. TYPES OF JOURNAL BEARINGS. Dr. Luis San Andrés © 2009 7

References consulted [1] Design of Journal Bearings for Rotating Machinery, P. Allaire & R.D. Flack, Proc. of the 10th

Turbomachinery Symposium, TAMU, pp. 25-45, 1981 [2] Fluid Film Bearing Fundamentals and Failure, F. Zeidan & B. Herbage, Proc. of the 20th

Turbomachinery Symposium, TAMU, pp. 161-186. 1991. [3] Fundamentals of Fluid Film Journal Bearing Operation and Modeling, M. He & J. Byrne, Proc. of

the 34th Turbomachinery Symposium, TAMU, pp. 155-176, 2005

Documents

NOTES 5 DYNAMICS OF A RIGID ROTOR FLUID FILM BEARING SYSTEMrotorlab.tamu.edu/me626/Notes_pdf/Notes05 rigid rotor on JBs 10.pdf · notes 5. dynamics of a rigid rotor-fluid film bearing